Verified Answer :

  • Identify the polynomial coefficients as \(1, -5, 0\) and the divisor value as \(2\).
  • Execute the synthetic division: bring down \(1\), multiply by \(2\) to get \(2\), add to \(-5\) to get \(-3\), multiply by \(2\) to get \(-6\), and add to \(0\) to get \(-6\).
  • Confirm the result using the Remainder Theorem: \(f(2) = 2^2 - 5(2) = 4 - 10 = -6\).
  • State the final remainder as \(\boxed{-6}\).

Explanation

  1. Understanding the Problem and Data Analysis

The problem asks for the remainder of a synthetic division with a divisor of \(2\) and a polynomial represented by coefficients \(1, -5, 0\), corresponding to \(f(x) = x^2 - 5x + 0 = x^2 - 5x\). Synthetic division is a shorthand method of polynomial division by a linear factor of the form \(x - c\). In this case, the divisor is represented by \(2\), which corresponds to \(c = 2\). We aim to find the remainder, which is the value obtained after completing the synthetic division steps.

  1. Performing the Synthetic Division

To perform the synthetic division, we follow these steps:

  1. Bring down the first coefficient, which is \(1\).
  2. Multiply this coefficient by the divisor: \(1 \times 2 = 2\).
  3. Add this result to the next coefficient: \(-5 + 2 = -3\).
  4. Multiply this new value by the divisor: \(-3 \times 2 = -6\).
  5. Add this result to the final coefficient: \(0 + (-6) = -6\).

The final value obtained, \(-6\), is the remainder of the synthetic division.

  1. Conclusion and Final Result

The synthetic division process yields a final result of \(-6\). This value is the remainder of the division of the polynomial \(x^2 - 5x\) by \(x - 2\). According to the Remainder Theorem, this is equivalent to evaluating the polynomial \(f(x) = x^2 - 5x\) at \(x = 2\): \(f(2) = 2^2 - 5(2) = 4 - 10 = -6\). This confirms our result. Thus, the remainder is \(\boxed{-6}\).

Examples

Synthetic division is a powerful tool in algebra for evaluating polynomials and finding their roots. Imagine you are modeling the trajectory of a ball, where the height is given by a quadratic equation. If you need to determine the height of the ball at a specific time (like at \(t=2\) seconds), you can use synthetic division as a quick alternative to plugging the value directly into the equation. This method is particularly useful when working with more complex polynomials, making calculations efficient and reducing the chance of arithmetic errors in real-world engineering or physics problems.

Related Questions